10 prediction of resilient modulus from soil index properties
Post on 22-Oct-2015
99 Views
Preview:
TRANSCRIPT
PREDICTION OF RESILIENT MODULUS FROM SOIL INDEX PROPERTIES
FINAL REPORT
By
K.P.George
Conducted by the
DEPARTMENT OF CIVIL ENGINEERING
THE UNIVERSITY OF MISSISSIPPI
In cooperation with
THE MISSISSIPPI DEPARTMENT OF TRANSPORTATION
And
U.S DEPARTMENT OF TRANSPORTATION FEDERAL HIGHWAY ADMINSTRATION
The University Of Mississippi
University, Mississippi November 2004
i
Technical Report Documentation Page
1.Report No. FHWA/MS-DOT-RD-04-172
2. Government Accession No.
3. Recipient’s Catalog No.
5. Report Date
August 2004 4. Title and Subtitle Final Report Resilient Modulus Prediction Employing Soil Index Properties 6. Performing Organization Code
7. Author(s) K.P.George
8. Performing Organization Report No.
MS-DOT-RD-04-172 10. Work Unit No. (TRAIS)
9. Performing Organization Name and Address Mississippi Department of Transportation Research Division P O Box 1850 Jackson MS 39215-1850
11. Contract or Grant No. State Study # 172
13. Type Report and Period Covered Final Report
12. Sponsoring Agency Name and Address Federal Highway Administration
14. Sponsoring Agency Code
15. Supplementary Notes
16. Abstract Subgrade soil characterization in terms of Resilient Modulus (MR) has become crucial for pavement design. For a new design, MR values are generally obtained by conducting repeated load triaxial tests on reconstituted/undisturbed cylindrical specimens. Because the test is complex and time-consuming, in-situ tests would be desirable if reliable correlation equations could be established. Alternately, MR can be obtained from correlation equations involving stress state and soil physical properties. Several empirical equations have been suggested to estimate the resilient modulus. The main focus of this study is to substantiate the predictability of the existing equations and evaluate the feasibility of using one or more of those equations in predicting resilient modulus of Mississippi soils. This study also documents different soil index properties that influence resilient modulus. Correlation equations developed by the Long Term Pavement Performance (LTPP), Minnesota Road Research Project, Georgia DOT, Carmichael and Stuart Drumm et al., Wyoming DOT, and Mississippi DOT are studied/analyzed in detail. Eight road (subgrade) sections from different districts are selected and soils tested (TP 46 Protocol) for MR in the laboratory. Other routine laboratory tests are conducted to determine physical properties of the soil. Validity of the correlation equations are addressed by comparing measured MR to predicted MR. In addition, variations expected in the predicted MR due to inherent variability in soil properties is studied by the method of point estimates. The results suggest that LTPP equations are suited for purposes of predicting resilient modulus of Mississippi subgrade soils. For fine-grain soils, even better predictions are realized with the Mississippi equation. A sensitivity study of those equations suggests that the top five soil index properties influencing MR include moisture content, degree of saturation, material passing #200 sieve, plasticity index and density. 17. Key Words Resilient Modulus, Subgrade, Prediction equations, Soil properties, Stress state
18. Distribution Statement Unclassified
19. Security Classif. (of this report) Unclassified
20. Security Classif. (of this page) Unclassified
21. No. of Pages 72
22. Price
Form DOT F 1700.7 (8-72)
ii
ACKNOWLEDGMENT This report includes the results of a study titled “Resilient Modulus Prediction Employing
Soil Index Properties”, conducted by the Department of Civil Engineering, The University of
Mississippi, in cooperation with the Mississippi Department of Transportation (MDOT), the U.S.
Department of Transportation, and Federal Highway Administration (FHWA). Funding of this
project by MDOT and FHWA is gratefully acknowledged.
The author wishes to thank Bill Barstis with MDOT’s Research Division for his technical
contribution to successful completion of this project,
Madan Gaddam and Khalid Desai, Research Assistants with the Department were the key
personnel from the University, conducting data analysis and providing support while preparing
the report. Their contributions are acknowledged.
DISCLAIMER
The opinions, findings and conclusions expressed in this report are those of the author
and not necessarily those of the Mississippi Department of Transportation or the Federal
Highway Administration. This report does not constitute a standard, specification or regulation.
iii
PREDICTION OF RESILIENT MODULUS FROM SOIL INDEX PROPERTIES:
A CRITICAL REVIEW
ABSTRACT
Subgrade soil characterization in terms of Resilient Modulus (MR) has become crucial for
pavement design. For a new design, MR values are generally obtained by conducting repeated
load triaxial tests on reconstituted/undisturbed cylindrical specimens. Because the test is complex
and time-consuming, in-situ tests would be desirable if reliable correlation equations could be
established. Alternately, MR can be obtained from correlation equations involving stress state and
soil physical properties. Several empirical equations have been suggested to estimate the resilient
modulus. The main focus of this study is to substantiate the predictability of the existing
equations and evaluate the feasibility of using one or more of those equations in predicting
resilient modulus of Mississippi soils. This study also documents different soil index properties
that influence resilient modulus.
Correlation equations developed by the Long Term Pavement Performance (LTPP),
Minnesota Road Research Project, Georgia DOT, Carmichael and Stuart, Drumm et al.,
Wyoming DOT, and Mississippi DOT are studied/analyzed in detail. Eight road (subgrade)
sections from different districts were selected, and soils tested (TP 46 Protocol) for MR in the
laboratory. Other routine laboratory tests were conducted to determine physical properties of the
soil. Validity of the correlation equations are addressed by comparing measured MR to predicted
MR. In addition, variations expected in the predicted MR due to inherent variability in soil
properties is studied by the method of point estimates. The results suggest that LTPP equations
are suited for purposes of predicting resilient modulus of Mississippi subgrade soils. For fine-
iv
grain soils, even better predictions are realized with the Mississippi equation.
A sensitivity study of those equations suggests that the top five soil index properties
influencing MR include moisture content, degree of saturation, material passing #200 sieve,
plasticity index and density.
v
TABLE OF CONTENTS
1. INTRODUCTION .................................................................................................................................... 1 1.1 BACKGROUND ....................................................................................................................................... 1 1.2 WHY THIS STUDY? ................................................................................................................................ 2 1.3 OBJECTIVE AND SCOPE:......................................................................................................................... 2
2. REVIEW OF LITERATURE.................................................................................................................. 4 2.1 INTRODUCTION ...................................................................................................................................... 4 2.2 WHY REPEATED LOAD TRIAXIAL TEST FOR DETERMINATION OF MR.................................................... 5 2.3 LABORATORY TEST TO DETERMINE RESILIENT MODULUS.................................................................... 6 2.4 FACTORS AFFECTING RESILIENT MODULUS .......................................................................................... 7 2.5 RESILIENT MODULUS BASED ON SINGLE SOIL PARAMETER .................................................................. 8 2.6 REGRESSION EQUATIONS FOR RESILIENT MODULUS BASED ON SOIL PROPERTIES AND STRESS STATE 9 2.7 RESILIENT MODULUS CONSTITUTIVE MODELS.................................................................................... 12 2.8 PREDICTION MODELS OF MR BASED ON CONSTITUTIVE EQUATION .................................................... 15 2.9 COMPARISON OF PREDICTIVE EQUATIONS FOR DETERMINATION OF MR (34) ...................................... 21 2.10 CRITIQUE OF EXPLANATORY VARIABLES FOR MR PREDICTION ......................................................... 22 2.11 SUMMARY ......................................................................................................................................... 22
3. EXPERIMENTAL WORK.................................................................................................................... 24 3.1 INTRODUCTION .................................................................................................................................... 24 3.2 LABORATORY TESTS ........................................................................................................................... 24
3.2.1 Routine Laboratory Tests............................................................................................................. 24 3.2.2 Laboratory Resilient Modulus Test.............................................................................................. 25
3.3 SUMMARY ........................................................................................................................................... 25 4. ANALYSIS AND DISCUSSION ........................................................................................................... 31
4.1 INTRODUCTION .................................................................................................................................... 31 4.2 RESILIENT MODULUS DETERMINATION............................................................................................... 31 4.3 PREDICTION OF MR EMPLOYING CORRELATION EQUATIONS............................................................... 32
4.3.1 Prediction of MR from LTPP Equations....................................................................................... 33 4.3.2 Prediction of MR from Georgia DOT Equations......................................................................... 33 4.3.3 Prediction of MR from Minnesota Equations ............................................................................... 34 4.3.4 Prediction of MR from Carmichael and Stuart Equations ........................................................... 35 4.3.5 Prediction of MR from Drumm’s Equation .................................................................................. 35 4.3.6 Prediction of MR from Wyoming Equations................................................................................. 36 4.3.7 Prediction of MR Employing Mississippi Equations .................................................................... 36 4.3.8 Comparison of Laboratory MR and Predicted MR from Various Models .................................... 37
4.4 PREDICTABILITY OF EQUATIONS UNDER UNCERTAINTIES ................................................................... 39 4.4.1 Method of Point Estimates........................................................................................................... 39 4.4.2 Variance in Model Prediction...................................................................................................... 40
4.5 MODEL VALIDATION ........................................................................................................................... 43 4.6 SENSITIVITY ANALYSIS OF MODELS.................................................................................................... 43 4.7 SUMMARY ........................................................................................................................................... 45
5. SUMMARY AND CONCLUSIONS ..................................................................................................... 56 5.1 SUMMARY ........................................................................................................................................... 56 5.2 CONCLUSIONS ..................................................................................................................................... 57 5.3 RECOMMENDATION/IMPLEMENTATION OF RESULTS ........................................................................... 57
6. REFERENCES ....................................................................................................................................... 59
7. APPENDIX ............................................................................................................................................. 63
vi
LIST OF TABLES
TABLE 3.1 TEST SECTION LOCATIONS AND PROCTOR TEST RESULTS OF BAG SAMPLES ................................. 27 TABLE 3.2 SOIL INDEX PROPERTIES OF BULK SAMPLES FROM VARIOUS SECTIONS ......................................... 27 TABLE 3.3 UNCONFINED COMPRESSIVE STRENGTH RESULTS FOR FINE-GRAIN SOILS ..................................... 28 TABLE 4.1 CONSTANTS (K-VALUES) FROM REGRESSION ANALYSIS OF RESILIENT MODULUS OF SUBGRADE SOILS
.................................................................................................................................................... 46 TABLE 4.2 MR VALUES CALCULATED FOR STRESS STATE, Σ1=7.4 PSI AND Σ3= 2 PSI ..................................... 47 TABLE 4.3 PREDICTION OF CONSTANTS (K-VALUES) AND MR FROM LTPP EQUATIONS................................ 48 TABLE 4.4 COMPARISON OF AVERAGE MR: (I) LABORATORY MR VS. PREDICTED MR FROM VARIOUS MODELS, (II)
VARIABILITY IN PREDICTION EMPLOYING POINT ESTIMATES (PE) METHODS ............................. 49 TABLE 4.5 PREDICTION OF CONSTANTS AND MR FROM GEORGIA DOT EQUATIONS ..................................... 50 TABLE 4.6 PREDICTION OF CONSTANTS (K-VALUES) AND MR FROM MINNESOTA ROAD EQUATIONS ........... 50 TABLE 4.7 COEFFICIENT OF VARIATION FOR SOIL ENGINEERING TESTS ....................................................... 50 TABLE 4.8 LIST OF SOIL PROPERTIES EMPLOYED IN MODEL BUILDING. ....................................................... 52 TABLE 4.9. RANK ORDER (BY COUNT) OF IMPORTANT VARIABLES.............................................................. 53 TABLE 4.10 MODEL VALIDATION BASED ON TWO CRITERIA ........................................................................ 53 TABLE 4.11 SENSITIVITY ANALYSIS (EFFECT OF RESPONSE VARIABLES ON MR PREDICTION) SILT SOILS #154 TABLE 4.12 SENSITIVITY ANALYSIS (EFFECT OF RESPONSE VARIABLES ON MR PREDICTION) CLAY SOILS # 455 TABLE 4.13 RANKING OF RESPONSE VARIABLES BASED ON SENSITIVITY..................................................... 55
LIST OF FIGURES FIGURE 3.1 RESILIENT MODULUS VS. DEVIATOR STRESS AT THREE CONFINING PRESSURES, SECTION #1, SAMPLE #1
.................................................................................................................................................. 28 FIGURE 3.2 RESILIENT MODULUS VS. DEVIATOR STRESS AT THREE CONFINING PRESSURES, SECTION #1, SAMPLE #2
.................................................................................................................................................. 29 FIGURE 3.3 RESILIENT MODULUS VS. DEVIATOR STRESS AT THREE CONFINING PRESSURES, SECTION #6, SAMPLE #31
.................................................................................................................................................. 29 FIGURE 3.4 RESILIENT MODULUS VS. DEVIATOR STRESS AT THREE CONFINING PRESSURES, SECTION #7, SAMPLE #1
.................................................................................................................................................. 30
1
CHAPTER 1
INTRODUCTION
1.1 Background
Characterizing subgrade soils in terms of resilient modulus (MR) is essential for pavement
design of both flexible and rigid pavements. The 1986 AASHTO guide for design of flexible
pavements (1) replaces soil support value (SSV) and recommends the use of MR for
characterizing the subgrade soil as it indicates a basic material property which can be used in
mechanistic analysis of multi-layered systems. MR attribute has been recognized widely for
characterizing materials in pavement design and evaluation. Resilient modulus is a measure or
estimate of the elastic modulus of the material at a given stress or temperature. Mathematically it
is expressed as the ratio of applied deviator stress to recoverable strain.
MR = σd / εr (1.1)
where, σd = Applied deviator stress
εr = Resilient strain.
MR is generally estimated directly in the laboratory using repeated load triaxial testing,
indirectly through correlation with other standard tests, or by back calculating from deflection
tests results. For a new design, MR is generally obtained by conducting repeated load triaxial tests
on reconstituted/undisturbed samples, according to harmonized test protocol, NCHRP1-28A (2).
Because tests are complex and time consuming, in-situ tests would be desirable if reliable
correlation could be established. Alternately, resilient modulus can be obtained from the
correlation equations involving stress state and soil physical properties.
The 1993 AASHTO Guide for Design of Pavement Structures: Appendix L (3), lists four
different approaches to determine a design resilient modulus value. The first approach is
2
laboratory testing, another approach is by Non-Destructive Testing (NDT) backcalculation, the
third approach consists of estimating resilient modulus from correlations with other properties,
and the last is from original design and construction data. In 1995, Darter, et al.(4), reported that
about 75 percent of the State Highway Agencies (SHAs) in the United States use either 1986 or
1993 versions of the AASHTO design guide. However, most of the agencies do not routinely
measure the MR in the laboratory, but estimate from experience or from other material or soil
properties; i.e., CBR, R-value or physical properties.
1.2 Why this Study?
Two types of correlation equations have been developed in predicting resilient modulus
from soil physical properties; they will be described in detail in the next chapter. Since several
equations are available from past studies, there is a need to substantiate the predictability of these
equations. Those equations, if proved to be valid, could serve a vital role in proposing a
preliminary pavement design for budgeting purposes. Final design can await completion of the
grading contract, followed by additional in-situ tests.
1.3 Objective and Scope:
As suggested in the 1993 AASHTO Guide, resilient modulus can also be predicted
directly from correlation equations involving soil index properties. Long Term Pavement
Performance (LTPP) (5), Minnesota Road Research project (6), Santha (7), Carmichael and
Stuart (8), Drumm (9), Farrar and Truner (10) and the Mississippi equation derived by Ashraf
and George (11) are of special interest in this study. A few other researchers also have
developed correlation equations to predict the resilient modulus from soil physical properties.
The primary objective of this study is to validate the existing equations cited in the
literature (5,6,7,8,9,10 and 11) and evaluate the feasibility of using one or more equations for
3
predicting the design resilient modulus of Mississippi soils. To achieve this objective, eight road
(subgrade) sections representing typical Mississippi soils were selected and tested in the
laboratory for MR in accordance with the AASHTO TP46 test protocol. In addition, routine
laboratory tests were conducted on the soils to determine the physical properties. Validation of
the equations is accomplished by comparing measured MR with the predicted MR. In addition,
expected variation in predicted values owing to inherent soil variability is investigated,
employing the Point Estimate (PE) method. Model sensitivity is examined by evaluating to what
extent each independent (response) variable effects the predicted MR value.
4
CHAPTER 2
REVIEW OF LITERATURE
2.1 Introduction
The objective of pavement design is to provide a structural and economical combination
of materials such that it serves the intended traffic volume in a given climate over the existing
soil conditions for a specified time interval. Traffic volume, environmental loads, and soil
strength determine the structural requirements of a pavement, and failure to characterize any of
them adversely affects the pavement performance. Traffic is estimated from present traffic and
traffic growth projections. Climatic conditions are incorporated in the design by accounting for
their effects on material properties. The subgrade may be characterized in the laboratory or by
field tests or both. It is essential that methods adopted to characterize reflect the actual
subgrade’s role in the pavement structure, and the frequency of the sampling should account for
spatial variation in the field. As noted by Yoder and Witczak (12) “all pavements derive their
ultimate support from the underlying subgrade: therefore, knowledge of basic soil mechanics is
essential.”
The AASHTO guide for the design of pavement structures, which was proposed in 1961
and then revised in 1972, characterized the subgrade in terms of soil support value (SSV). SSV
has a scale ranging from 1 to 10, with a value of 3 representing the natural soil at the Road Test.
In the revised 1986 AASHTO guide, the road bed resilient modulus, MR, was selected to replace
the SSV, used in the previous editions of the Guide, for the following reasons:
1. It indicates a basic material property, which can be used in mechanistic analysis of multi-
layered systems for predicting roughness, cracking, rutting, faulting, etc.
5
2. MR has been recognized internationally as a method for characterizing materials for use
in pavement design and evaluation.
3. Methods for determination of MR are described in AASHTO Test Method T274-82 and
others, the latest being Harmonized Test, NCHRP 1-28A.
4. Techniques are available for estimating the resilient properties of various materials in-
place by non-destructive tests.
2.2 Why Repeated Load Triaxial Test for Determination of MR
Since the pavement materials are subjected to a series of distinct load pulses, a laboratory
test duplicating this condition is desirable. The repeated load type test has been used for many
years to simulate vehicle loading. In this test, cylindrical specimens of soil are subjected to a
series of load pulses applied with a distinct rest period, simulating the stresses caused by multiple
wheels moving over the pavement. A constant all-around confining pressure applied on the
specimen simulates the lateral stresses caused by the overburden pressure and applied wheel
load. The total resilient (recoverable) axial deformation response of the specimen to the stress
pulses measured is used to calculate the resilient modulus of the material. Cited below are two
reasons favoring the use of repeated load triaxial test for determination of MR (13).
• In the triaxial test, predetermined principal stresses σ1 and σ3 are applied to the specimen;
therefore, the stress conditions within the specimen on any plane are defined throughout
the test. The stress conditions applied are, in fact, those which occur when an isolated
wheel load is applied to the pavement directly above the element of material simulated in
the test.
• Axial, radial, and volumetric strains can all be measured in the triaxial test.
For about the last 35 years the repeated load triaxial compression test has been the basic test
6
procedure to evaluate resilient modulus of cohesive and granular materials for pavement design
applications.
2.3 Laboratory Test to Determine Resilient Modulus
The 1986 AASHTO Guide has stipulated and the 2002 Guide reaffirmed, that the MR be
the parameter for characterizing the subgrade. Responding to the need, AASHTO T278-82
laboratory test was proposed to describe the behavior of pavement materials subjected to moving
traffic. In 1991, AASHTO modified the T278-82 testing procedure in terms of sample
conditioning, load magnitude, and load application. With the revised test designation it changed
to TP292-92I. Later, TP46-94, a “harmonized” MR test protocol, was proposed in the NCHRP 1-
28A study; and the latest is the P46 proposed by LTPP. In this study, samples,
undisturbed/reconstituted, are subjected to the repeated load triaxial test in accordance with the
AASHTO (TP-46) protocol to determine the resilient modulus. For undisturbed samples, Shelby
tube sampling is relied upon. Reconstituted samples are molded in the laboratory to obtain
desired density and moisture content representative of the field.
The sample in the repeated load triaxial test is subjected to a combination of three
confining stresses and five deviator stresses, thus yielding 15 resilient modulus values for each
sample. Now, a constitutive model comprising MR-stress relation is chosen, describing the
resilient property of the material. This model is then fitted to the data of each sample so the MR
for a desired stress state can be obtained.
Despite several improvements made over the years, Seed et al. (14) cited the following
uncertainties as well as limitations associated with the test procedure.
7
1. The laboratory resilient modulus sample is not completely representative of in-situ
conditions because of sample disturbance and differences in aggregate orientation, moisture
content and level of compaction.
2. Inherent equipment flaws make it difficult to simulate the state of stress of material in-situ.
3. Inherent instrumentation flaws create uncertainty in the measurement of sample
deformation.
4. Lack of uniform equipment, calibration, and verification procedures may lead to differences
between the laboratories and within a given laboratory.
Overall, these issues have kept MR testing from achieving general acceptance by the researchers
as well as user agencies.
2.4 Factors Affecting Resilient Modulus
The resilient modulus of fine-grain soils is not a constant stiffness property (15) but
depends upon various factors like load state or stress state, which includes the deviator and
confining stress, soil type and its structure, which primarily depends on compaction method and
compaction effort of a new subgrade. Previous studies show that deviator stress is more
significant than confining stress for fine-grain soils. Resilient modulus is found to increase with a
decrease in moisture content and an increase in density, and decrease with an increase in deviator
stress.
For coarse-grain soils, MR is primarily influenced by the stress state, degree of saturation
and compactive effort (density). Research (10, 16) has shown that MR increases with increasing
confining stress. Studies have also indicated that there is a critical degree of saturation near 80-
85 percent, above which granular material becomes unstable and undergoes degradation rapidly
under repeated loading. Lekarp et al. (17) noted, and other researchers concur that MR of
8
granular materials increases with increasing confining stress and sum of principal stresses,
otherwise known as bulk stress (θ), and slightly increases with deviator stress.
2.5 Resilient Modulus Based on Single Soil Parameter
Simple correlation equations have been reported to predict MR from standard California
Bearing Ratio (CBR), R value, and soil physical properties. Heukelom and Klomp (18) reported
correlation between the Corps of Engineers CBR value using dynamic compaction and the in-
situ resilient modulus of soil.
MR (psi) = 1500 CBR (2.1)
The data used for developing this equation ranged from 750 to 3000 times CBR. Equation (2.1)
has been extensively used by design agencies and researchers for fine grained soils with a soaked
CBR of 10 or less.
The Georgia Department of Transportation tested a number of cohesionless soils in
repeated load triaxial test following the AASHTO procedure (19). The objective was to create a
database so that the resilient modulus can be predicted. A typical equation for medium clay sand
follows:
MR (psi) = 3116 (CBR)a (2.2)
where a = 0.4779707
The results showed no significant change in resilient modulus as long as soils are within ± 1.5 %
of optimum.
Though CBR is widely used to characterize subgrade soils, it is a measure of shear
strength, which is not necessarily expected to correlate with modulus or stiffness. Thompson and
Robnett (20) reported weak correlation between CBR and resilient modulus of Illinois soils.
Besides, CBR-based relationships do not recognize the stress dependence on modulus (21) and
9
are, therefore, critiqued to be inadequate.
Similar relationships were developed by the Asphalt Institute (22) which related R value
to resilient modulus. Their equation is,
MR (psi) = A + B (R value) (2.3)
where, A = 772 to 1155;
B = 369 to 555; and
R value = Stabilometer value, lbs
Yeh and Su (23) of the Colorado Department of Highways tested the resilient properties
of Colorado soils with the objective of establishing a correlation between resilient modulus and
stabilometer R value. Triaxial modulus was determined adopting a procedure different from
AASHTO T274. The equation finally derived between the MR and R value is as follows:
MR (psi) = 3500 + 125 (R value) (2.4)
The fundamental problem with empirical relationships developed to correlate resilient
modulus to soil parameters such as CBR or R value is that those tests themselves are pretty much
empirical. Whereas resilient modulus is a mechanistic parameter and dependent on a host of soil
index properties and stress state.
2.6 Regression Equations for Resilient Modulus Based on Soil Properties and Stress State
Carmichael and Stuart in 1985 (8) studied the resilient properties of soils with the
objective of developing correlation equations for predicting subgrade modulus from basic soil
tests. The Highway Research Information Service (HRIS) database was searched to compile the
necessary data for correlation analysis. Regression studies were made for individual soil types
according to the Unified Soil Classification system. Two models were developed, one for fine-
grain soils and another for coarse-grain soils. Equation 2.5 presents the model for coarse-grain
10
soils.
Log MR = 0.523–0.025(wc) + 0.544(log θ) + 0.173(SM) + 0.197(GR) (2.5)
where, MR = Resilient Modulus, ksi;
wc = moisture content, %;
θ = bulk stress (σ1+σ2+σ3 ), psi;
SM = 1 for SM soils (Unified Soil Classification)
= 0 otherwise; and
GR = 1 for GR soils (GM, GW, GC or GP)
= 0 otherwise.
A different equation was derived for fine-grain soils:
MR = 37.431–0.4566(PI)–0.6179(wc)–0.1424(P200)+0.1791(σ3)–0.3248(σd)+36.722
(CH) +17.097 (MH). (2.6)
where, PI = plasticity index, %;
P200 = percentage passing #200 sieve;
σ3 = confining stress, psi;
σd = deviator stress, psi;
CH = 1 for CH soil
= 0 otherwise (for MH, ML or CL soil); and
MH = 1 for MH soil
= 0 otherwise (for CH, ML or CL soil).
Drumm et al. (9) conducted a resilient modulus study on cohesive soils, employing
AASHTO test specifications. The authors tried to establish a simple procedure for modulus
testing. Their result showed that unconfined compressive strength, qu, is a better property to
11
predict MR. Accordingly, they correlated the soil index properties and the initial tangent modulus
obtained from unconfined compression test to the resilient modulus. A statistical model
developed with a nonlinear relationship between resilient modulus and the deviator stress
follows:
MR (ksi) = a' + b' σd for σd > 0 (2.7) σd
where, a' = 318.2 + 0.337 (qu) +0.73(%Clay)+2.26(PI)–0.915(γs)–2.19(S)
–0.304(P200); (2.8)
b' = 2.10+0.00039(1/a)+0.104(qu)+0.09(LL)–0.10 (P200); (2.9)
qu = unconfined compressive strength, psi;
1/a = initial tangent modulus, psi, obtained from unconfined compression tests;
%Clay = percent clay;
LL = liquid limit, %;
S = degree of saturation; and
γs = dry density, pcf.
It was concluded that a similar relationship could be established for soils other than those
investigated and might be helpful to agencies that lack the capability for complex repeated load
testing.
Two regression models, to predict resilient modulus, were developed, employing thirteen
fine-grain Wyoming soils (10). In the first model, R value is the primary response variable, and
in the other soils index properties and stress state, namely, σ3. The latter equation, investigated
in this study, follows:
MR = 34280 – 359 S% - 325 σd + 236 σ3 + 86 PI + 107 P200 (2.10)
Note that the authors of the Wyoming study observed that resilient modulus is negatively
12
correlated with degree of saturation and positively associated with PI and P200.
Ashraf and George (11) investigated the relevance of soil index properties in predicting
resilient modulus of Mississippi soils. Two equations, herein referred to as Mississippi equations,
were proposed, one for fine-grain soil and another for coarse-grain soil. The former equation
developed using 12 soils from Mississippi, had been substantiated with eight other soils, also
from Mississippi. The two models are presented here:
Fine-grain soil:
MR (MPa) = 16.75((LL/wc γdr)2.06 + (P200/100) -0.59) (2.11)
Coarse-grain soil:
MR (MPa) = 307.4 (γdr/ wc )0.86 (P200/log cu) -0.46 (2.12)
where, γdr = dry density/maximum dry density; and
cu = uniformity coefficient
2.7 Resilient Modulus Constitutive Models
The concept of resilient modulus has been used to explain the nonlinear stress-strain
characteristics of subgrade soils. During the past two decades, several constitutive models have
been proposed by many researchers for modeling resilient moduli of soils and aggregates. No
stress or deformation analysis can be meaningful unless a correct constitutive equation
describing the actual behavior of the material has been used in the analysis. In 1963, Dunlap
(24) suggested the following relationship for presenting resilient modulus data:
MR = k1 (σ3/Pa) k2 (2.13)
where, k1, k2 = regression coefficients obtained from regression analysis,
Pa = reference pressure (atmospheric pressure); and
σ3 = confining stress.
13
However, this relationship does not consider the effect of deviator stress on resilient modulus.
Seed et al. (25) suggested a relation where resilient modulus is a function of bulk stress
(θ), also known as the K–θ model. This model, generally adopted for granular soils, uses θ as the
main attribute in the model.
MR = k1 (θ/Pa) k2 (2.14)
Where, θ = bulk stress (σ1+σ2+σ3 ).
The main drawback of this model is that it does not account for shear stresses and shear
strains developed during loading and is, therefore, applicable only in the range of low-strain
values (26). Brown and Pappin (27) noted that this model does not properly handle volumetric
strains or dilative behavior of soils. Moreover, this model potentially provides the same resilient
properties for the same bulk stress input. This shows that the model does not incorporate the
realistic responses of confining and deviator stresses on resilient properties.
Moossazadeh and Witczak (28) proposed a relation known as the deviator stress model
recommended for reporting cohesive soil results, known as K-σd model. It uses deviator stress as
the main and only attribute of the model.
MR = k1 (σd /Pa) k2 (2.15)
where, σd = deviator stress (σ1 - σ3).
Though this model does not consider the effect of confining stress on resilient modulus,
for clay soils, this aspect is still considered insignificant since cohesive soils derive their overall
strength mainly from cohesion rather than from frictional characteristics. This modeling
approach is perhaps adequate for cohesive soils found at shallow depths, but for soils found at
greater depth under high traffic loads, it is necessary to include confining stress in addition to
deviator stress (23).
14
May and Witczak (29) and LTPP (30) proposed a model to describe the nonlinear
behavior revealed in the repeated load triaxial test. This model considers the effects of shear
stress, confining stress and the deviator stress with the model formulated in terms of bulk and
deviator stress.
MR = k1 Pa (θ /Pa) k2 (σd /Pa) k3 (2.16)
Uzan in 1992 introduced octahedral shear stress in place of deviator stress in Equation
(2.16), which provided a better explanation for the stress state of the material, in which the
normal and shear stress change during loading. The proposed model is known as the k1- k3 model
or universal model. The universality of this model stems from its ability to conceptually
represent all types of soils from pure cohesive soils to non-cohesive soils.
MR = k1 Pa (θ /Pa) k2 (τoct/Pa) k3 (2.17)
where, τoct = ( ) ( ) ( )( ) 212
132
322
2131 σσσσσσ −+−+−
The coefficients k1, k2, and k3 are constants, dependent on material type and physical
properties, and are obtained from regression analysis. Since coefficient k1 is proportional to
Young’s modulus, it should always be positive as MR can never be negative. The coefficient k2
should be positive, because increasing the volumetric stress produces stiffening or hardening of
the material, yielding higher modulus. The coefficient k3 should be negative because an increase
in the shear stress softens the material, thereby yielding lower modulus. If nonlinear property
coefficients k2 and k3 are set to zero, then the model can be simplified as linear elastic. If k3 is
zero, the behavior could be non-linear hardening and if k2 is zero, the behavior is non-linear
softening.
Various modified versions of the universal equation are currently used to
predict/calculate MR. A modified version, proposed in Long Term Pavement Performance
15
research (31), has been adopted in this study. The model takes the following form:
Log(MR/Pa) = k1 + k2 Log(θ/Pa) + k3 Log(τoct /Pa) + k4 [Log(τoct /Pa)]2 (2.18) In the
above expression, an additional second order term of octahedral shear stress has been introduced,
since there is a reasonably strong trend for TP46 results to be more nonlinear in octahedral shear
stress (31).
2.8 Prediction Models of MR Based on Constitutive Equation
Most of the State Highway Agencies in the United States do not routinely measure MR in
the laboratory but estimate the design MR either from experience or from other material
properties. The potential benefit of estimating MR from soil physical properties is that the
seasonal variations in the MR can be determined from seasonal changes in the material’s
properties; however, the effect of stress sensitivity is not captured. In order to capture the effects
of stress sensitivity and physical properties on design MR, Von Quintos and Killingsworth (32),
Dai et al.(6), Santha (7) and Mohammad et al. (33), among others, have developed prediction
equations for MR by regressing the coefficients of selected constitutive equations relating them to
soil physical properties.
Researchers in the past have developed relationships between the soil properties and the
regression coefficients (k1, k2, k3) of the constitutive equation. Those relations that have good
statistics were generally confined to specific soil types (7). Other studies that used a wide range
of soils and conditions resulted in poor correlations (32). Included is a list of studies to predict
MR based on the constitutive relations and physical properties:
1. Long Term Pavement Performance Study (5);
2. Georgia Department of Transportation Research study (7);
3. Minnesota Road Research Project (6); and
16
4. Louisiana Study (33).
The primary soil properties which influence resilient modulus are moisture content,
density and percent passing # 200 sieve, and liquid limit and plasticity index. It was observed
that MR increases with an increase in density and decreases with an increase in moisture content
above optimum. Hence, these soil index properties were invariably used to frame correlation
equations. Other properties like compressive strength, degree of saturation, percent clay, percent
silt and CBR were also used in a few equations.
The LTPP-FHWA study program (5) is a comprehensive review of the resilient modulus
test data measured on pavement materials and soils recovered from the LTPP test sections. The
MR data was reviewed in detail to identify anomalies or potential errors in the database, and the
effect of test variables such as the test and sampling procedures on the resilient modulus. The
resilient modulus data was further investigated to evaluate relationships between MR and the
physical properties of the unbound materials and soils. Equations for each base and soil type
were developed to calculate MR at a specific stress state from physical properties of the base
materials and soils using nonlinear regression optimization techniques.
The constitutive equation used is of the form:
MR = k1 Pa (θ /Pa) k2 [(τoct/Pa) +1] k3 (2.19)
The regression constants k1, k2 and k3 are material-specific, as listed in the following equations,
2.20-2.28.
• For coarse-grained sand soils, the k1 – k3 constants are described as follows:
k1 = 3.2868 – 0.0412 P3/8 +0.0267 P4 + 0.0137 (%Clay) + 0.0083 LL – 0.0379 wopt –
0.0004 γs (2.20)
k2 = 0.5670 + 0.0045 P3/8 – 2.98x10-5 P4 – 0.0043 (%Silt) – 0.0102 (%Clay) – 0.0041
17
LL + 0.0014 wopt – 3.41x10-5 γs – 0.4582 (γs / γopt ) + 0.1779 (wc/wopt) (2.21)
k3 = -3.5677 + 0.1142 P3/8 – 0.0839 P4 - 0.1249 P200 + 0.1030 (%Silt) + 0.1191
(%Clay) – 0.0069LL – 0.0103 wopt – 0.0017 γs + 4.3177(γs / γopt ) –
1.1095 (wc/wopt ). (2.22)
• Fine-grain silt soils:
k1 = 1.0480 + 0.0177 (%Clay) + 0.0279 PI – 0.0370 wc (2.23)
k2 = 0.5097 – 0.0286 PI (2.24)
k3 = -0.2218 + 0.0047 (%Silt) + 0.0849 PI – 0.1399 wc (2.25)
• Fine-grain clay soils:
k1 = 1.3577 + 0.0106 (%Clay) – 0.0437 wc (2.26)
k2 = 0.5193 – 0.0073 P4 + 0.0095 P40 - 0.0027 P200 – 0.003 LL – 0.0049 wopt (2.27)
k3 = 1.4258 – 0.0288 P4 +0.0303 P40 – 0.0521 P200 + 0.0251 (%Silt) + 0.0535 LL –
0.0672 wopt – 0.0026 γopt + 0.0025 γs – 0.6055 (wc / wopt ) (2.28)
where, MR = Resilient Modulus, MPa;
P3/8 = percentage passing sieve #3/8;
P4 = percentage passing #4 sieve;
P40 = percentage passing #40 sieve;
wc = moisture content of the specimen, %;
wopt = optimum moisture content of the soil, %;
γs = dry density of the sample, kg/m3 ; and
γopt = optimum dry density, kg/m3 .
The resilient modulus test results of the laboratory reconstituted samples were
exclusively used in developing the correlation equations, because the samples taken for
18
measuring the soil physical properties by Shelby tube were at different depths. The primary
result from these studies is that the resilient modulus can be reasonably predicted from the
physical properties included in the LTPP database.
Santha (7) of the Georgia Department of Transportation compared two well known
constitutive models (bulk stress model and universal model) in modeling granular subgrade soils,
concluding that MR of granular soils is better described by the universal model. Also studied
were the effects of material and physical properties of subgrade soils on the resilient moduli.
Subgrade soil samples were reconstituted in the laboratory and tested for MR according to the
AASHTO T 274-82 procedure. Results showed that the k-parameters in the constitutive equation
can be estimated using the soil physical properties, and the values of the k-parameter vary over a
wide range of cohesive and granular soils. From the study of 14 cohesive soils and 15 granular
soils correlation equations were developed. A multiple correlation analysis approach was used to
obtain the relationships among k-parameters (dependent variable) and soil properties such as
percent passing #40 sieve (P40), percent passing #60 sieve (P60), percent clay (%Clay), percent
silt (%Silt), percent swell (%SW), percent shrinkage (SH), maximum dry density (γd), optimum
moisture content (wopt), California Bearing Ratio (CBR), sample moisture content (wc), sample
compaction (COMP) and percent saturation (SATU). Two correlation equations were developed,
one for granular soils and the other for cohesive soils, and they are of the following form.
For granular soils:
MR = k1 Pa (θ /Pa) k2 (σd /Pa) k3 (2.29)
where, Log k1=3.479–0.07wc+0.24wc ratio+3.681COMP+0.011 %Silt+0.006 %Clay–
0.025SW–0.039 γs+0.004(SW2/ %Clay)+0.003(γs 2/ P40); (2.30)
k2=6.044 – 0.053wopt – 2.076COMP + 0.0053SATU – 0.0056%Clay +
19
0.0088SW–0.0069SH – 0.027 γs + 0.012 CBR + 0.003 (SW2 / %Clay)
– 0.31(SW+SH) / %Clay; and (2.31)
k3= 3.752–0.068wc + 0.309wc ratio– 0.006 %Silt + 0.0053 %Clay
+0.026SH–0.033γs–0.0009(SW2/%Clay)+0.00004(SATU2/SH)–
0.0026(CBR*SH). (2.32)
For cohesive soils:
MR = k1 Pa (σd / Pa )k3 (2.33)
where, Log k1 =19.813–0.045 wopt–0.131wc–9.171 COMP + 00337 %Silt + 0.015 LL
– 0.016 PI – 0.021 SW – 0.052 γs + 0.00001 (P40*SATU); (2.34)
k3 = 10.274 – 0.097 wopt – 1.06 wc ratio – 3.471 COMP + 0.0088 P40 –
0.0087PI + 0.014 SH – 0.046 γs ; and (2.35)
wc ratio = moisture content of specimen / optimum moisture.
Dai et al. (6), in an attempt to compare the two well known constitutive models
(Universal model and deviator stress model) in describing subgrade soil resilient behavior, and to
study the effects of material properties on the MR, selected Shelby tube samples from six
different pavement sections of the Minnesota Road Research project. Repeated load triaxial tests
were conducted on the soil specimens to determine MR at Minnesota DOT laboratories along
with some other soil property tests. Resilient modulus test data is used in the regression analysis
to develop correlation equations between the model constants (k1, k2, k3) and the soil physical
properties. The constitutive equation developed to predict the resilient modulus for a stress state
with soil physical properties takes the following form:
MR = k1 θ k2 σd k3 (2.36)
where, k1 = 5770.8–520.98 (γs)0.5–3941.8(wc)0.5+33.1PI–36.62 LL–17.93 P200 (2.37)
20
k2 = -5.334+0.000316(γs)3+9.686(wc)–0.054PI+0.046LL+0.022 P200 (2.38)
k3 = 409.9 –306.18 (γs)0.1–82.63(wc)+0.033 PI+0.138S–0.041LL (2.39)
γs = dry density of soil specimen, kN/m3 ; and
S = saturation, %.
However, the relationships presented in this study were based on soils with physical index
properties of a narrow range. Therefore, the predictability of the model is suspect.
In order to validate the octahedral stress state model in characterizing resilient modulus,
Mohammad et al. (33), selected eight different soils representing major soil types in Louisiana
and tested for MR in the laboratory. Additional analysis was performed to develop correlations
between the model parameters and soil properties. Multiple linear regression analysis was
performed between the model constants of the constitutive equation and the basic soil properties.
The correlation equations developed are as follows:
MR = k1 Pa (σoct / Pa ) k2 ( τoct / Pa ) k3 (2.40)
where, k1, k2, k3 are the regression constants listed in the following equations 2.41-2.43.
Log k1 = - 0.679+ 0.0922 wc+0.00559 γs+3.54 (γs / γopt)+2.47 wc ratio+0.00676 LL +
0.0116 PL+0.022 (%sand)+0.0182 (%silt) (2.41)
Log k2 = - 0.887+0.0044wc+0.00934 γs+0.264(γs / γopt)+ 0.305 wc ratio+0.00877 LL+
0.00665 PL+0.0116 (%sand)+0.00429 (%silt) (2.42)
Log k3 = - 0.638+0.00252wc+0.00207γs +0.61 (γs / γopt)+0.152 wc ratio+0.00049 LL+
0.00416 PL+0.00311 (%sand)+0.00143 (%silt). (2.43)
where, Pa = atmospheric pressure, psi;
σoct= octahedral normal stress, (σ1 + σ2 + σ3 )/3 , psi;
τoct = octahedral shear stresses, psi;
21
γs = dry density, kN/m3; and
PL = plastic limit, %;
It appeared from the analysis that model constants for resilient modulus were mainly governed
by density, moisture content, liquid limit, and plastic limit, the same soil attributes widely
employed in several other models. Based on the study, it was recommended that the models be
used for the prediction of resilient properties of Louisiana subgrade soils.
Preliminary analysis of the model revealed that the above equations were unsuitable in
predicting resilient modulus of Mississippi soils. Upon contacting the authors with the result,
however, they referred to certain ongoing work to improve the model; which was not available to
the researcher in time for this report.
2.9 Comparison of Predictive Equations for Determination of MR
With numerous equations proposed over the years, a comparison of their predictability
was undertaken in a recent study by Kyatham et al. (34). They compared primarily three
equations: the bilinear model by Thompson and Robnett (20) and Drum et al. (9), and Farrar and
Turner (10). The former two equations predict the breakpoint resilient modulus whereas the latter
predicts MR directly for a given stress state. Breakpoint modulus refers to the modulus at which
the slope of MR versus deviator stress changes. Soil test results from four states – Illinois,
Wyoming, Tennessee and New Jersey –have been discussed and analyzed in detail to determine
if any of those predictive equations are universally applicable. Based on the analysis it was
concluded that there is no universally available predictive equation to estimate resilient modulus.
The study suggested that Universal Model (Eq. 2.16) is suitable for determining resilient
modulus as a function of confining pressure and deviator stress, but the constants should be
determined at a stress range of interest.
22
2.10 Critique of Explanatory Variables for MR Prediction
Soil index properties commonly used in developing the correlation equations in the order
of importance are material passing # 200 sieve, Atterberg limits (LL, PI), moisture content, and
dry density. Though used in several models, no definite trend can be seen between resilient
modulus and material passing # 200 sieve. A cursory examination of LTPP equations suggests,
however, that MR attains a peak value in the range of 40 to 60 percent material passing # 200
sieve. From a majority of the equations it can be seen that, MR increases with an increase in PI.
Though PI is an important factor, its effect on MR is inconsistent with the general soil mechanics
principles namely, the higher the PI the less stable the soil is. A soil with a PI value in the range
of 10 to 20 percent is considered satisfactory. Intuitively, MR should decrease with an increase in
moisture above optimum, however, different equations show different trends. In equations, for
example 2.32-2.38, MR increases with an increase in the moisture. MR increases with the percent
clay, in the range of 10 to 40 percent, beyond which it decreases. Regarding the effects of density
on MR, the results are inconclusive because in several equations, (for example, 2.21, 2.22, 2.31,
2.32 and 2.34) MR decreases with increase in density. From a physical point of view, one would
expect MR to increase with the density.
In general, LTPP study (5), proposes the following broad conclusions. Liquid limit,
plasticity index, and material passing #200 sieve are important for the lower strength materials,
while a measure of moisture content and density are important for the higher strength materials.
Percent silt is important for all soil groups, excluding gravel soils.
2.11 Summary
Resilient modulus of subgrade soil is an important material property, a requisite
parameter to input in the pavement design equation, generally determined in the laboratory by
23
performing a repeated load triaxial test (AASHTO TP 46) procedure. Because the test is complex
and time consuming several user agencies in the United States and abroad now estimate design
resilient modulus from correlation equations developed from soil physical properties. This
chapter presents various forms of correlation equations including constitutive models and the
importance of soil properties in their formulation. A cursory study of the equations suggest that
soil index properties such as material passing #200 sieve, Atterberg limits, moisture and dry
density significantly affect MR. Due in part to nonlinear behavior of soil, stress state becomes an
important parameter as well.
24
CHAPTER 3
EXPERIMENTAL WORK
3.1 Introduction
By comparing predicted MR with laboratory MR only, validity of equations is appraised.
With the objective of compiling laboratory MR, subgrade soil samples collected from different
locations in Mississippi are classified and tested for resilient modulus in the MDOT laboratory,
in accordance with the AASHTO TP46 protocol. Employing the soil index properties and a
realistic stress state, resilient modulus is predicted using the correlation equations cited in the
previous chapter and compared with measured resilient modulus. A summary of the tests
conducted along with results of each soil is presented in the ensuing sections.
3.2 Laboratory Tests
The soils tested in this study were selected to provide a general representation of typical
subgrade soils in Mississippi. Eight different subgrade soils from nine different sections were
tested. All of the eight soil materials have been used recently in subgrade construction. These test
sections were selected in connection with a study investigating the use of a Falling Weight
Deflectometer for subgrade characterization (35).
Composite bag samples were collected from each section for routine laboratory tests and
resilient modulus tests as well. A summary of section locations is presented in column 2 of Table
3.1. Also listed in Table 3.1 are the Standard Proctor test results.
3.2.1 Routine Laboratory Tests
The eight subgrade soil samples were classified into fine-grain and coarse-grain soils
according to AASHTO classification. Laboratory tests performed to classify the soils are the
25
Particle size distribution test (AASHTO T88-90), Liquid limit test (T89-90), Plastic limit test
(T90-87), and Standard Proctor test (T99-90). An unconfined compressive strength test was
performed on all the fine-grain soils in accordance with AASHTO T208-90. Soil index
properties are listed in Table 3.2. Since Drumm’s equation required unconfined compressive
strength and initial tangent modulus as inputs they were determined as presented in Table 3.3.
3.2.2 Laboratory Resilient Modulus Test
Making use of the bulk material from each section, three cylindrical samples 2.8 inch (71
mm) diameter by 5.8 inch (147 mm) length were molded at the target density (i.e. the maximum
dry density) and optimum moisture content, as listed in Table 3.1. These samples were prepared
in three layers in a split mold, each layer receiving 25 blows with a tamping rod 5/8 inches (16
mm) diameter. The final compaction was accomplished by a compressive load of the order of
5000 lbs. Wrapped with cellophane wrap, they were stored in a humidity room for 5 days and
then tested in the Repeated Load Triaxial machine in accordance with the AASTHO TP46 test
protocol. The tests were conducted using the MDOT repeated load triaxial machine, supplied by
Industrial Process Control (IPC), Borona, Australia. The load sequence and the combinations are
presented in Appendix A. Axial deformation of the specimen is recorded by two externally
mounted Linear Variable Differential Transducers (LVDT). The average of the resilient modulus
values of the last five loading cycles of the 100 cycle sequence yields the requisite resilient
modulus. Typical plots of laboratory MR test results of reconstituted samples related to deviator
stress are presented in Figure 3.1 to 3.4, the former two figures for a fine-grain and the latter two
for a coarse grain soil.
3.3 Summary
A detailed discussion of the laboratory tests performed on the bulk samples is presented.
26
Summary of the physical properties of the samples are presented as well. Detailed discussion and
analysis of the test results will be the topic of the following chapter.
27
Table 3.1 Test section locations and Proctor test results of Bag samples Section # County / Road Section
Length (ft) Optimum Moisture
Content (%)
Dry Density ( lb/ft3)
1 Montgomery /US 82 W 200 13.8 115.2
2 Coahama / US 61 N 500 14.1 113.7 3 Coahama / US 61 N 200 12.9 116.2 4 Montgomery / US 82 W 200 13.8 115.5 6 Hinds / Norell W. 200 17.8 105.6 7 Wayne / US 45 N 200 11.0 118.0
8/9 Wayne / US 45 N 200 12.0 118.9 10 Madison county/Nissan west
parkway 200 18.6 106.1
1 ft = 0.305m; 1 lb/ft3 = 0.157 kN/m3; Table 3.2 Soil index properties of bulk samples from various sections
Classification Sec-tion
#
Liquid Limit (%)
Plasticity Index (%)
Passing # 200 sieve (%)
Passing # 40 sieve (%)
Clay(%)
Silt (%)
AASH-TO
U S C
1 22.3 6.1 55.0 NA 10.6 44.5 A4 CL 2 27.0 8.0 56.0 NA 14.2 41.8 A4 CL-ML 3 25.0 7.0 40.0 NA 10.8 45.2 A4 SM-SC 4 28.1 12.4 60.0 90 12.3 48.1 A6 CL 6 37.2 13.1 96.0 99 19.3 78.7 A6 CL 7 20.5 1.0 28.0 NA 3.2 25.4 A2-4 SM
8/9 24.4 4.9 42.0 NA 9.0 33.1 A4 CL-ML 10 35.8 13.3 98.0 99 18.9 79.1 A6 CL
28
Table 3.3 Unconfined compressive strength results for fine-grain soils Section # Unconfined compressive strength (psi) Initial tangent modulus, (psi)
1 15.4 2300 2 26.6 2500 3 18.5 1400 4 17.4 2400 6 26.9 3350
8/9 20.7 2400 10 25.4 2300
1 psi = 6.89 kPa
40
50
60
70
80
90
100
110
120
0 10 20 30 40 50 60 70Deviator Stress, kPa
Res
ilien
t Mod
ulus
, MPa
37.7Kpa25.8 kpa13.4 kpa
Figure 3.1 Resilient Modulus vs. Deviator Stress at Three Confining Pressures,
Section #1, Sample #1; 1 kPa = 0.15 psi
29
40
50
60
70
80
90
100
110
0 10 20 30 40 50 60 70Deviator Stress, KPa
Res
ilien
t Mod
ulus
, MPa
36.9 kpa24.9 kpa12.7 kpa
Figure 3.2 Resilient Modulus vs. Deviator Stress at Three Confining Pressures,
Section #1, Sample #2; 1 kPa = 0.15 psi
70
80
90
100
110
120
130
0 10 20 30 40 50 60 70Deviator Stress, kPa
Res
ilien
t Mod
ulus
, MPa
37.7Kpa25.7 kpa13.4 kpa
Figure 3.3 Resilient Modulus vs. Deviator Stress at Three Confining Pressures, Section
#6, Sample #31; kPa = 0.15 psi
30
60
70
80
90
100
110
120
130
140
0 10 20 30 40 50 60 70Deviator Stress, kPa
Res
ilien
t Mod
ulus
, MPa
37.7Kpa25.8 kpa13.6 kpa
Figure 3.4 Resilient Modulus vs. Deviator Stress at Three Confining Pressures, Section
#7, Sample #1; 1 kPa = 0.15 psi
31
CHAPTER 4
ANALYSIS AND DISCUSSION
4.1 Introduction
The primary objective of this study is to explore the predictability of the correlation
equations reported in previous research studies, for example, LTPP program, Georgia DOT,
Minnesota Road Research project, Carmichael and Stuart, Drumm et al., Wyoming and Ashraf
and George. This chapter presents the data and analysis comparing measured resilient modulus to
predicted resilient modulus from the aforementioned correlation equations. Also presented in this
chapter is the expected variability in predicting resilient modulus owing to inherent soil property
variations. Model sensitivity results are also presented seeking the significance of response
variables in predicting MR. A discussion on the feasibility of using existing correlation equations
is also presented at the end of this chapter.
4.2 Resilient Modulus Determination
Three reconstituted samples of 2.8 inch (71mm) diameter and 5.8 inch (147 mm) height
from the bag samples collected from each section are molded and tested for MR in accordance
with TP46 protocol. The cylindrical samples are subjected to 15 stress combinations (three
confining stresses and five deviator stresses) yielding 15 MR values. Principal stresses, bulk
stress (θ), and octahedral shear stress (τ) are calculated for each stress combination. The equation
(2.18) is then fitted for each set of data expressing MR as a function of θ and τoct. Regression
constants (k1 to k4) of the reconstituted samples are listed in Table 4.1.
Regression constants (k1 to k4 values) indicate that MR increases with an increase in bulk
stress as suggested by positive values of k2, and decreases with an increase in octahedral shear
32
stress as k3 is negative. This relation holds well as long as one of the terms is kept constant and
the other is changed, but whether a soil exhibits stress hardening or softening under simultaneous
change in bulk and shear stress state would depend on the nature of the stress state and the
resulting change in bulk and octahedral stresses. Hence, it is difficult to predict how a soil
behaves under changing stress states, but can only be analyzed by substituting the corresponding
values of shear and bulk stresses.
Since MR is a function of the stress state, for calculating a representative MR for
correlation, an appropriate stress state has to be selected because the AASHTO Guide does not
mandate one. A stress state pertaining to the actual condition (overburden stress) of soil in its
final location, including the stress generated by a standard axle load could be an appropriate
scenario. Relying on the results of Thompson and Robnett (20), Elliot (36) suggested using a
zero confining pressure and a 6 psi (41.6 kPa) deviator stress when selecting an MR value from
laboratory test data. In the field, however, subgrade has to sustain the overburden of pavement
layers, in addition to the standard 18-kip axle load. In-situ stress in a typical subgrade includes
stresses due to a 4500 lb (20 kN) wheel load at a tire pressure of 100psi (690 kPa) and
overburden pressure as well. Stress analysis by KENLAYER (37) yielded a stress state of 7.4 psi
(51 kPa) vertical stress and 2 psi (14 kPa) lateral compressive stress. This stress combination is
used to calculate the representative MR of each sample, and the average of the three samples of a
section is taken as a representative MR of the section for the given stress state. The individual MR
values of the samples of each section and their average are presented in Table 4.2.
4.3 Prediction of MR Employing Correlation Equations
Resilient moduli of eight subgrade soils are predicted employing the correlation
equations cited in the previous chapter.
33
4.3.1 Prediction of MR from LTPP Equations
LTPP correlation equations (2.20-2.28) were employed to predict the resilient modulus of
eight subgrade soils based on soil index properties. Regression coefficients calculated from
LTPP equations and the resulting MR are presented in Table 4.3. Note that stress state of σ1 = 7.4
psi (51 kPa) and σ 2 = σ 3 = 2 psi (14 kPa) is the input in calculating MR values.
Since coefficient k1 in the equation 2.19 is proportional to the Young’s modulus, it must
always be positive as MR cannot be negative. Increasing the bulk stress should produce stiffening
or hardening of the material, resulting in higher modulus. So the constant k2 of the constitutive
equation should be positive. Since increasing shear stress should produce a softening of the
material, values of k3 should be negative. From Table 4.3, it can be observed that coefficient k1
and k2 are positive and k3 negative, just what is expected of typical soils. A simple comparison of
predicted MR (column 3 of Table 4) and laboratory values (column 2 of Table 4) suggests fair
agreement in five out of eight soils. Treating soil 3 laboratory MR with reservation, soils 6 and 10
are the ones showing large discrepancy in prediction. A cursory examination of the Universal
model, having two exponential coefficients, suggests that significant errors could be introduced
in predicted values of MR even for moderate changes in k2 and k3.
4.3.2 Prediction of MR from Georgia DOT Equations
Equations 2.33, 2.34 and 2.35 for fine-grain soil were solved inputting the soil properties
of seven Mississippi soils, and the results are presented in Table 4.5. Comparing the laboratory
and predicted values, it is clear that the model severely over predicts MR values. Difficulties
encountered in using the equation are, first, a suspected typographical error in the equation could
not be verified by the authors, and second, percent swell (SW) for the seven soils had to be
estimated from PI data.
34
The question persisting is why the equation over predicts MR values (compare columns 2
and 5 of Table 4.4). A cursory examination of k1 and k3 values suggest that k3 fluctuates
significantly from soil to soil causing corresponding large variations in MR values (see Table 4-
5). Second, the Georgia DOT equation was developed by relying on 14 fine-grain soils of
average clay content and PI of 38% and 18% respectively, both significantly larger than those for
which predictions are made. Average clay content and PI of Mississippi soils are 14% and 9%,
respectively. Attempting to predict beyond the inference space of the model seems to be the
primary reason for the over prediction. Third, the Georgia equation denies “deductive result”.
For example, MR decreases with compaction ratio (note the large regression coefficient) as well
as the effect of density contradictory to an intuitive result that MR should increase with density.
Though this requirement is not a requisite condition, a physically intuitive model stands a better
chance of being “robust”. In other words, the validity of the equation is limited only to Georgia
soils.
With the necessity to estimate even more variables in the coarse-grain equations, the
predicted value failed to match the laboratory MR value, therefore, those results are not presented
in this report for brevity.
4.3.3 Prediction of MR from Minnesota Equations
Equations (2.36-2.39) were employed to predict the resilient modulus of the eight
Mississippi soils. Regression coefficients (k-values) derived and MR predicted from the
correlation equations are presented in Table 4.6.
It can be observed from Table 4.6, that negative resilient modulus is predicted for the
sections 6 and 10 because the constant k1 predicted was negative, due in part to excessive
material passing the #200 sieve. Equation 2.37 could not predict positive k1 values for soils 6 and
35
10, as the equation was developed employing soils having a narrow range of material passing the
#200 sieve (57 to 68 percent). Since inference space of the model is limited, soils 6 and 10, with
more than 95 percent material passing the #200 sieve, could not predict reasonable values. It was
mentioned earlier that constant k2 should be positive and k3 negative. But for sections 1, 2, 3, 4,
7 and 8/9 the k2-value predicted was negative, which could be attributed to low moisture content
and low liquid limit of Mississippi soils relative to Minnesota soils. The positive value of k3 for
sections 1, 2 and 4 is attributed to the low liquid limit of the soils. With no valid estimated k-
values, resilient modulus predictions for the eight subgrade soils are highly unsatisfactory. In
other words, the correlation equation developed in the Minnesota Road project could not
reasonably predict the resilient moduli of Mississippi subgrade soils, implying that the model
does not satisfy the validation criterion.
4.3.4 Prediction of MR from Carmichael and Stuart Equations
Carmichael and Stuart equations (2.5-2.6), which heavily depend on soil index properties,
are used in predicting the resilient modulus of eight subgrade soils. Table 4.4 lists the MR values.
In-order to compute MR, a deviator stress of 5.4 psi (37 kPa) and lateral stress of 2.0 psi (14 kPa)
were employed. As can be seen from the Table 4.4 (Column 9), resilient moduli for sections 6
and10 were predicted low because of the large amount of material passing the #200 sieve. An
observation is in order here that moisture content and percent passing the #200 sieve are more
significant than the deviator stress and the confining stress.
4.3.5 Prediction of MR from Drumm’s Equation
Resilient modulus of eight subgrade soils has been predicted with regression models
(equations 2.7-2.9) developed by Drumm et al. They are reported in column 11 of Table 4.4.
36
The constant a' estimated from the prediction equation for all the soils is within the range of the
values used in developing the model. For sections 1, 4, 6 &10, predicted b' values, however, are
out of range compared to the values used in developing the equation. Low compressive strength
and low clay content of the samples could have resulted in this outcome.
Note that resilient modulus is 2 to 3 times larger than the initial tangent modulus obtained
from unconfined compression tests. Though no conclusive proof can be offered, this result seems
quantitatively reasonable.
4.3.6 Prediction of MR from Wyoming Equations
Resilient modulus of 7 soils predicted by Eq. 2.10 can be seen in Table 4.4 (Column 13).
The predicted values are substantially lower than the TP 46 values except in two soils namely, #6
and #10, both with nearly 98% passing the #200 sieve. Note that the predicted MR values are
relatively small in soils #3 and #8/9, coincidentally with small amounts of material passing the
#200 sieve, namely 40% and 42%, respectively. Accordingly, it is presumed that P200 plays a
major role in the prediction process. The 13 Wyoming soils employed in developing the model
had an average P200 value of 73% (P200 range from 43% to 89%), suggesting that the inference
space of the Wyoming equation is limited, curtailing its predictability in Mississippi soils of
widely varying fines content. In other words, much like for other equations, the validity of the
Wyoming equation is also suspect.
4.3.7 Prediction of MR Employing Mississippi Equations
Despite being simple and concise, the equation predicts moduli values close to the
laboratory values in 6 out of 7 soils. Coarse-grain soil prediction is not as good, perhaps affected
by the explanatory variable, uniformity coefficient (Cu). The uniformity coefficient, the ratio of
37
D60 over D10, is considered less precise than the other soil index properties. Note also that only
one sample is available to test the validity of the equation. It would appear that an additional
explanatory variable, for example, confining stress could well improve the predictability of the
equation.
4.3.8 Comparison of Laboratory MR and Predicted MR from Various Models
Table 4.4 presents a comparison of the laboratory MR with resilient modulus predicted
from various models. Assuming laboratory resilient modulus represents the true modulus of the
material, it is compared and critiqued with the predicted values.
From Table 4.4, it can be seen that the MR predicted by LTPP equations for fine-grain silt
soils (sections 1, 2 and 8/9) is comparable, though somewhat smaller in section 2 and 8/9.
Section 3 is an exception. For sections 4, 6 and 10, which are fine-grain clay soils, very low
modulus is predicted primarily due to excessive silt (≈80%), and relatively small amounts of
clay. And for section 7, which is a coarse-grain sand soil, again MR is under predicted, which can
be attributed to low clay content (3%). It can be seen from Table 4.4 that an average deviation of
MR of seven soils (excluding section 3, which is an outlier) from LTPP equations is within 19%
percent of the measured MR, which is encouraging.
Georgia DOT equations severely over predict MR values of Mississippi soils. As cited in
a previous section, over prediction could be attributed to large variations of k3. Also, the 14
Georgia soils, from which the equation was developed, are uniformly high plasticity clay soils
where as Mississippi soils are predominantly silty.
Minnesota Road research equations could not satisfactorily predict the constants (k-
values) for the Mississippi subgrade soils. Resilient modulus is over predicted for some sections
and under predicted for others. Since soils from one project, with a very narrow range of
38
properties were used in developing the prediction equations, they are not versatile enough to
satisfactorily predict the resilient modulus of Mississippi soils. Simply put, the validity of this
equation in predicting MR of soils of other texture is questionable.
Resilient modulus predictions from Carmichael and Stuart-equations yielded high
modulus for some sections and low modulus for others. Again very low modulus predicted for
sections 6 and 10 is primarily due to excessive fines (≈96%). Note that material passing the #200
sieve of sample data in developing the equation ranged from 60 to 90 percent only. And, the high
modulus predicted for section 1,2,3, & 8/9 could be attributed to (i) material passing the #200
sieve being relatively small and (ii) the plasticity index also low relative to the PI range of soils
employed in developing the equations. Only the coarse soil MR (section 7) is predicted
satisfactorily.
Drumm’s equation satisfactorily predicts the resilient modulus of fine-grain soils
(sections 1,2, and 6). On average, all of the MR predictions are within 15% of the measured MR.
One drawback of the Drumm’s model is that it relies on variables such as initial tangent modulus
and unconfined compressive strength. The fact that the initial tangent modulus is relatively
difficult to estimate could affect the predictions. Note that Drumm’s equation is especially suited
for fine-grain soils; no such equation is available for coarse-grain soils.
A majority of the predictions by the Wyoming equation is substantially lower than the
laboratory MR, except in soils #6 and #10. As discussed in a previous section, the inference space
of the equation comprised of A-7-6 soils with large amount of P200 (73%). This P200 is relatively
larger than those found in most Mississippi soils. Also, note that the contribution of P200 to MR
value in the equation is substantial. In other words, using this equation for Mississippi soils
entails extrapolation of the equation. The Wyoming equation, therefore, is judged to be invalid
39
for Mississippi soils.
The Mississippi equation predicts six out of seven fine soils within 10 Mpa, which is
considered a satisfactory match. It should be pointed out that the equation was developed from
an independent set of soils different from those whose MR values are predicted, and compared.
4.4 Predictability of Equations under Uncertainties
In all of the correlation equations identified in the literature MR turns out to be a function
of soil index properties and stress states. Once a grading job is completed, the subgrade soil
could show significant spatial variation, resulting in randomness and uncertainty. Therefore, it is
important to estimate how this variability would affect the predicted MR values. The problem
then boils down to estimating the variability in predicted MR as the independent variables change
over a reasonable range. This problem can be studied either by method of point estimates (38) or
Taylor’s series expansion method.
4.4.1 Method of Point Estimates
The method of Point Estimates (PE) facilitates computations for the first two moments
(i.e. mean and variance) of a dependant variable in terms of the first two moments of the
independent variables. Approximate formulas for the moment’s calculation can be obtained
from a Taylor series expansion of the function about the first moment of the random variables.
However, due to excessive restrictions imposed on the function (existence and continuity of the
first few derivatives) and the requirement to compute the derivatives, the moment’s calculation
turns out to be difficult. These difficulties can be overcome through the use of the method of
point estimates. Equations required to perform the required calculations are presented in
Appendix B.
In the correlation equations 2.19, 2.29, 2.36 and 2.40, MR is expressed as a function of
40
stress state. The constants (k values), which are exponents of the stress state are, in turn,
functions of soil index properties. The first two moments of the k values are calculated from the
mean and variance of the independent variables (soil physical properties). Subsequently, mean
and variance of MR are, calculated based on the first two moments of the k-values. Thus, the
variations expected in predicting MR due to inherent variability in soil index properties are
quantified. Soil properties for each section were determined by performing routine laboratory
tests. With mean values obtained from laboratory tests, coefficient of variation is adapted from a
list of values in reference 39. The suggested coefficient of variation of each independent variable
is listed in Table 4.7. Since the complexity of PE analysis increases exponentially, the number of
input properties with inherent variability is limited to four in each case.
4.4.2 Variance in Model Prediction
By introducing variation in soil physical properties, expected variation in resilient
modulus prediction is calculated employing the method of point estimates. From a list of
independent variables included in column 3 of Table 4.8, three or four variables are chosen,
which are assumed to vary +/- one standard deviation (SD). As can be seen from the table a
variety of soil properties appear in those seven models investigated. Table 4.9 is prepared to rank
the response variables in the order of their importance. For example, P200 is shown to be the
most-often adopted variable, in six out of seven models. In addition to mean MR values, Table
4.4 lists the PE-based mean and coefficient of variation (CV) that are computed with each
equation.
Assuming inherent variability in four variables, the mean values calculated from LTPP
equations are reported in Table 4.4. As expected, the PE mean values (column 4 of Table 4.4),
are practically the same, and rightly so, as the mean values (computed by direct substitution of
41
the independent variables in the equation, column 3 of Table 4.4). Note that the predicted mean
MR values are smaller than the laboratory values, excluding section #3 which is deemed to be an
outlier. In general, a relatively small coefficient of variation is an indication that inherent
variations in the independent variables, i.e. soil index variables, would have minimal effect on
the predicted values. The fact that the average CV of 16% for 9 soils, therefore, reinforces the
previous observation that LTPP equations are capable of predicting MR from soil physical
properties.
PE mean values of Georgia DOT equations, when introducing variation in four soil
properties are listed in column 6 of Table 4.4. Not only the predicted mean values, but also the
CVs are relatively large. Accordingly, the validity of the Georgia DOT equation, in predicting
MR of Mississippi soils, is suspect.
PE analysis of Minnesota equations, assuming variability in four variables resulted in
unacceptably large mean values and coefficients by variation. Those results are not reported here
for brevity ruling out its use for predicting MR of Mississippi Soils.
The Georgia DOT and Minnesota equations both resulting in unrealistic mean and
coefficient of variation suggest an apparent weakness of the Universal model, i.e., expressing MR
as power functions of θ (volumetric stress) and τ, σd or σ3. The modeling entails deriving first k1,
k2 …., followed by another regression analysis where by those constants are expressed as
function of soil properties. The constant k1 and powers k2 and k3 could cause the equations to
result in unrealistic MR values. For example, k1 had become negative for soils #6 and #10, in the
case of the Minnesota equation. Therefore, PE mean values become negative. Another scenario
would be that the mean and standard deviations could blow up or even become negative should
k2 and/or k3 equations are not robust. Since k2 and k3 are exponents (typically k2 is positive and
42
k3 negative), small change in k2 and/or k3 could make large difference in predicted MR, which is
believed to be the reason for “wild” predictions of Minnesota and Georgia DOT equations. Note
that LTPP equations are not vulnerable to such large swings owing primarily to the robustness of
k1-, k2- and k3- equations.
Table 4.4 (Column 10) presents the mean and coefficient of variations when inherent
variability in index properties are introduced in the Carmichael equation. The PE mean values
are again very close to the predicted mean values with average soil properties, and coefficients of
variation of all of the soils are reasonably low, except in soils #6 and #10. Despite relatively
small CVs the agreement between measured (TP 46) and predicted MR values is less than
satisfactory. The one coarse soil (soil 7) investigated shows satisfactory agreement, however.
The means and coefficients of variation obtained in predicting MR from Drumm’s
equation are presented in column 12 of Table 4.4. Not only are the MR values under predicted
(excluding sections 2 and 3), but the CVs are relatively large as well. Large CV means large
swings in the predicted value with relatively minor variations in the basic soil property. Put
differently, when using Drumm’s equation, there is a good chance of predicting very high or
very low MR depending on the accuracy of input values.
Including variability in three independent variables and making PE calculations with the
Wyoming equation, not only are the mean MR predictions unsatisfactory, but the CVs are also
substantially large. As pointed out in previous sections, because the Wyoming equation had its
root in a database of fine-grain soils of nearly identical texture, it can hardly be valid for other
soils, in this case, Mississippi soils.
PE mean values of the Mississippi equation with inherent variability introduced in three
input parameters agree well with the mean values and TP 46 values as well. The CV of the
43
predicted MR values, however, are relatively large. Note that the predicted MR values in six out
of seven soils agree with the laboratory values.
4.5 Model Validation
The question now arises; which model(s) satisfies validation criteria? Validation refers to
the process to confirm that the proposed model can produce robust and accurate predictions for
cases other than those used in model development and/or calibration. Two criteria are employed
to accomplish this task. First, the predicted value shall agree with the laboratory measured value
within +/- 20% range. This 20% range is chosen with due consideration to the typical variability
observed in the laboratory MR values. Second, the inherent variability in input variables shall
not generate a large enough coefficient of variation, (not exceeding 20%). Table 4.10 lists to
what extent each criterion is satisfied for all of the six models. Respectively, one coarse- and
seven fine-soils are included.
Though none of the equations entirely meet the stated criteria, two equations stand out, in
regard to satisfying the two criteria. The LTPP equations present a strong showing in so far as
variability in predictions, and Mississippi equations satisfying the equality criterion (see Table
4.10). Specifically, the variability is satisfied by LTPP equations in one coarse soil and seven
fine soils. That is, in all of the soils the variability does not exceed 20%. And, Mississippi
equation predictions agree in six out of seven fine soils, and the only one coarse soil prediction
fails the 20% criterion. LTPP equation is successful due in part to them being derived from a
large database of soils collected and tested nationwide. The specialty of Mississippi equation is
that it is derived from Mississippi soils.
4.6 Sensitivity Analysis of Models
The sensitivity analysis examines the effect of each independent (response) variable on
44
the predicted MR value, in contrast to the PE analysis that evaluates the inherent variability
collectively of all of the independent variables. Independent variables include soil index
properties and compaction attributes, and stress state variables. Since soil properties are likely to
vary temporally and spatially, with them beyond designer’s control, sensitivity study seeks how
each of them effects MR prediction.
The methodology of a sensitivity study entails changing the mean value of each
independent variable by +/- one standard deviation and calculating the corresponding change in
the predicted MR value. The mean value of each independent variable is nothing but its
measured value, for example, material passing the #200 sieve determined by sieve analysis.
Typical values listed in Table 4.7 are the coefficient of variation adopted for each soil
index/compaction/strength property. One silty soil (soil 1), as well as a clay soil (soil 4) is
studied by changing each independent variable by plus or minus one standard deviation and
predicting MR. Tables 4.11 and 4.12 list those MR values for soils 1 and 4, respectively. Having
determined that the prediction of Minnesota and Georgia models are not satisfactory, a
sensitivity study of those models is not attempted. Accordingly, the results of only the remaining
five models are reported here.
Two criteria have been adopted to rank the sensitivity of response variables; namely,
“significance”, and “acceptable trend”. Explanatory variable is said to be significant when
predicted change (MR in this case) exceeds 10% with a change of one standard deviation of the
variable in question. The second criterion is that the predicted MR change shall be intuitively
satisfying, meaning that the trend of predicted change is physically meaningful. Based on those
two criteria, water content or its surrogate percent saturation appears to be the most significant
soil property, followed by material passing the #200 sieve and plasticity index in that order (see
45
Table 4.13). Density explicitly appears only in two equations: LTPP and Drumm. Its significance
in the LTPP fine grain soil equations is negligible, where as it turns out to be significant in the
Drumm equation. The trend of increasing density giving rise to decreasing resilient modulus,
however, is suspect, negating its role in those two equations. A density ratio, appearing in the
Mississippi equation, moderately effects MR prediction in fine soil. In coarse soil, density,
however, has a major role in the prediction. The sensitivity study and ensuing results should
guide the design engineer in paying special attention to those soil properties that are shown to be
significant.
Concluding, the sensitivity study not only singles out significant variables, but also
reveals the trend of MR accompanying the variation of each variable. For example, resilient
modulus is shown to be negatively correlated with water content/ degree of saturation, positively
associated with percent passing the #200 sieve, and a mixed trend with PI. Though only two
equations include soil density explicitly, the result that MR decreasing with increase in density is
suspect.
4.7 Summary
Resilient modulus is predicted for eight Mississippi subgrade soils employing selected
correlation equations, and compared with the laboratory resilient modulus. Comparisons reveal
that the Mississippi equation reasonably predicts MR of typical Mississippi fine-grain soils. A
statistical study of variance expected in calculating MR confirms that LTPP model predictions
are less prone to uncertainties that may arise from inherent variations in soil properties. A
sensitivity study of the soil properties in predicting MR has been conducted, choosing three soil
properties. Moisture content, percent passing the #200 sieve and PI are the most significant in
MR prediction.
46
Table 4.1 Constants (k-values) from Regression Analysis of Resilient Modulus Expressed by Equation 2.18 Section # Sample # k1 k2 k3 k4
1 1 2.448 0.616 -0.630 -0.073 2 2.496 0.471 -0.580 -0.064 3 2.433 0.606 -0.712 -0.140 2 1 2.655 0.258 -0.515 -0.151 2 2.607 0.230 -0.741 -0.323 3 2.536 0.381 -0.897 -0.413 3 1 2.551 0.470 -0.250 0.025 2 2.476 0.495 -0.431 -0.074 3 2.502 0.463 -0.363 -0.040 4 1 2.604 0.383 -0.734 -0.255 2 2.648 0.277 -0.823 -0.358 3 2.649 0.229 -0.730 -0.297 6 1 2.610 0.279 -0.670 -0.275 2 2.723 0.256 -0.494 -0.199 3 2.752 0.252 -0.469 -0.169 7 1 2.736 0.491 -0.331 -0.045 2 2.726 0.459 -0.384 -0.081 3 2.725 0.436 -0.398 -0.084
8/9 1 2.594 0.448 -0.831 -0.319 2 2.761 0.351 -0.521 -0.159 3 2.734 0.355 -0.657 -0.237
10 1 2.521 0.290 -0.734 -0.289 2 2.565 0.278 -0.707 -0.287 3 2.530 0.290 -0.778 -0.302
47
Table 4.2 MR Values Calculated for Stress State, σ1=7.4 psi and σ3= 2 psi
Section # Sample # MR, MPa Average MR, MPa 1 64.0 2 68.6
1
3 65.5
66.5
1 82.8 2 88.2
2
3 84.5
85.8
1 49.5 2 49.7
3
3 49.4
49.8
1 91.3 2 105.8
4
3 98.6
99.2
1 82.8 2 88.0
6
3 93.7
88.7
1 79.3 2 81.7
7
3 83.5
82.0
1 95.8 2 103.4
8/9
3 111.4
104.3
1 73.7 2 78.2
10
3 79.9
77.9
1 MPa = 0.15 ksi
48
Table 4.3 Prediction of Constants (k-values) and MR from LTPP Equations Section # Sample # k1 k2 k3 Predicted MR
(MPa) Average MR
(MPa) 1 0.903 0.335 -1.398 64.8 2 0.911 0.335 -1.370 65.6
1
3 0.907 0.335 -1.384 65.2
65.6
1 1.019 0.281 -1.249 75.8 2 1.015 0.281 -1.263 75.4
2
3 1.011 0.281 -1.277 74.9
75.8
1 0.968 0.310 -1.178 73.0 2 0.965 0.310 -1.192 72.0
3
3 0.972 0.310 -1.164 72.9
72.9
1 0.880 0.377 -0.874 68.3 2 0.885 0.376 -0.879 68.6
4
3 0.893 0.376 -0.864 69.4
70.1
1 0.780 0.344 -1.478 55.1 2 0.798 0.344 -1.444 56.7
6
3 0.798 0.344 -1.442 56.7
57.4
1 0.877 0.456 -1.286 62.5 2 0.877 0.454 -1.275 62.6
7
3 0.876 0.452 -1.263 62.7
64.7
1 0.989 0.338 -1.082 76.6 2 0.997 0.338 -1.054 77.5
8/9
3 1.000 0.338 -1.040 78.0
75.9
1 0.710 0.335 -1.704 48.5 2 0.723 0.334 -1.702 49.4
10
3 0.719 0.334 -1.704 49.1
51.9
1 MPa= 0.15 ksi
49
Table 4.4 Comparison of Average MR: (i) Laboratory MR vs. Predicted MR from Various Models, (ii) Variability in Prediction Employing Point Estimates method.
PE PE PE PE PE PE PEMean, Mpa
Mean, Mpa
Mean, Mpa
Mean, Mpa
Mean, Mpa
Mean, Mpa
Mean, Mpa
CV CV CV CV CV CV CV65.7 1037.9 191324.0 117.0 56.8 40.6 70.712% 97% 182% 12% 40% 52% 24%75.9 956.1 1854562.0 109.2 85.2 46.5 90.013% 95% 20% 14% 28% 45% 27%73.0 2193.6 26633.5 132.5 84.4 37.9 96.912% 68% 169% 10% 26% 53% 25%70.1 1628.2 336846.0 91.0 81.4 46.3 97.918% 101% 210% 20% 31% 47% 28%58.7 26280.3 -390000000.0 36.4 75.2 71.7 96.526% 110% -199% 65% 37% 30% 30%63.0 75.3 63.230% 13% 20%70.7 575.3 103260.2 144.0 76.2 37.1 103.17% 83% 168% 8% 29% 55% 26%53.6 24235.0 -490000000.0 30.4 59.8 69.2 84.027% 121% -197% 79% 46% 33% 29%
Drumm WyomingTP 46 Mean, MPa
LTPP Minnesota MississippiMean, MPa
Mean, MPa
Mean, MPa
Mean, MPa
Mean, MPa
Mean, MPa
Mean, MPa
Georgia Carmichael
116.2 56.8 40.61 66.0 65.6 260.8 68.9
2 85.2 75.8 233.0625.3 109.9 85.2 46.5
653.3
87.4
3 49.6 72.9 66.3917.6 132.7 84.4 37.9 94.2
73.2 46.3
76.2
4 98.6 70.1 252.3961.4
81.5 64.8
95.1
6 88.2 57.4 -34878.010693.9 37.1 75.1 75.5
90.7
27.9 59.8
93.6
7
8/9 103.5 75.9 35.4428.8 145.0
10 77.3 51.9 -28167.07954.4 73.9 81.5
37.1 100.2
Soil/ Section
#
Resilient Modulus
62.975.8
1 MPa = 0.15 ksi
50
Table 4.5 Prediction of Constants and MR from Georgia DOT Equations
Section # k1 k3 Mean MR (MPa)
TP 46 Mean (MPa)
1 3.836 0.0123 653.3 66.0 2 3.818 0.0165 625.3 85.2 3 3.986 0.327 917.6 49.6 4 3.936 -0.151 961.4 98.6 6 5.05 0.005 10693.9 88.2
8/9 3.655 0.325 428.8 103.5 10 4.875 -0.099 1954.4 77.3
1 MPa= 0.15 ksi
Table 4.6 Prediction of Constants (k-values) and MR from Minnesota Road Equations Section # Sample # k1 k2 k3 Predicted MR
(MPa) Average MR
(MPa) 1 502.9 -0.252 0.130 267.7 2 511.7 -0.262 0.122 254.0
1
3 507.4 -0.258 0.127 260.8
260.8
1 385.8 -0.167 0.056 228.3 2 381.5 -0.162 0.059 232.8
2
3 377.9 -0.161 0.066 237.7
233.0
1 744.8 -0.540 -0.018 66.2 2 739.8 -0.533 -0.013 68.9
3
3 749.2 -0.543 -0.026 63.6
66.3
1 387.1 -0.154 0.074 257.9 2 389.7 -0.150 0.065 256.2
4
3 398.8 -0.162 0.058 242.9
252.3
1 -665.4 0.945 -0.045 -34918 2 -651.9 0.927 -0.017 -35021
6
3 -652.5 0.930 -0.023 -34693
-34878
1 1005.4 -0.753 -0.216 17.2 2 1010.6 -0.759 -0.228 16.1
7
3 1015.0 -0.760 -0.247 15.1
16.1
1 733.5 -0.437 -0.292 37.8 2 743.2 -0.444 -0.324 33.2
8/9
3 737.7 -0.438 -0.311 35.4
35.4
1 -708.9 1.047 -0.254 -27288 2 -696.2 1.021 -0.199 -29130
10
3 -702.3 1.037 -0.231 -28080
-28167
1 MPa= 0.15 ksi
Table 4.7 Coefficient of Variation for Soil Engineering Tests (adapted from reference 39)
51
Sl.No Soil Property / Variable Coefficient of
Variation 1 Clay % 25 2 Silt % 25 3 P40 % 10 4 P200 % 15 5 Ws % 15 6 LL % 10 7 PI % 30 8 Density 5 9 Saturation 10 10 P3/8 25 11 P4 25 12 Cu 25 13 Unconfined Comp Strength 40
52
Table 4.8 List of Soil Properties Employed in Model Building. (The Last Column Lists the Variables Which are Assumed to Vary). Model Soil Texture List of Variables Selected Variables
Coarse-grain P3/8, P4, %Clay, LL, wopt, γs,
%Silt, γopt, wc, P200
%Clay, LL, wopt, %Silt,
γopt, P200
Fine-grain % Clay, PI, wc, % Silt % Clay, PI, wc, % Silt
LTTP
Fine-grain
clay
% Clay, wc, P4, P40, P200, LL,
wopt, % Silt, LL, γopt, γs
% Clay, wc, P40, P200,
LL, wopt, % Silt, LL,
Georgia Coarse-grain wopt, wc, COMP, %Silt, LL, PI,
SW, SH, P40, γs , S
wc, %Silt, PI, S
Minnesota Fine-grain γs, wc, PI, LL, P200,S γs, wc, PI, LL, P200,S
Coarse-grain wc, Soil Class wc, Carmichael
Fine-grain PI, wc, P200, Soil Class PI, wc, P200
Drumm Fine-grain qu, % Clay, PI, γs, S, wc, P200,
LL
qu, % Clay, PI, S, wc,
P200, LL
Wyoming Fine-grain PI, S, P200 PI, S, P200
Coarse-grain wc, γd, P200,cu wc, γd, P200 Mississippi
Fine-grain LL, wc, γd, γopt, P200 LL, wc, γd, γopt, P200
53
Table 4.9. Rank Order (By Count) of Important Variables
Sl. No. Soil Variable Rank Order
1 Passing # 200 (6/7)
2 PI (6/7)
3 Moisture (5/7)
4 LL (5/7)
5 Density (4/7)
Table 4.10 Model Validation Based on Two Criteria
Sl.No Model Agreement with TP46 (+/- 20%)
Coefficient of Variation
(20 %) Comments
1 2 3 4 5 6
LTTP (C) (F) Georgia (F) Carmichael (C) (F) Drumm (F) Wyoming (F) Mississippi (C) (F)
(1/1)a 2/7 0/7 1/1 1/7 3/7 2/7 0/1 6/7
1/1 7/7 0/7 1/1 5/7 0/7 0/7 1/1 1/7
Satisfactory Satisfactory Satisfactory Satisfactory
C = Coarse-grain soil
F = Fine-grain soil
a = Number of Soils satisfying each criterion out of total number tested
54
Table 4.11 Sensitivity Analysis (Effect of Response Variables on MR Prediction). Silt Soil #1 MR
+
= MR calculated with response variable increased by one standard deviation
MR- = MR calculated with response variable decreased by one standard deviation
1 MPa= 0.15 ksi
Resilient Modulus, MPa LTPP Carmichael Drumm Wyoming Mississippi
Response Variable
MR+ MR
- MR+ MR
- MR+ MR
- MR+ MR
- MR+ MR
- %Clay 69.0 62.2 - - 58.9 54.5 - - - -
%Silt 66.2 65.2 - - - - - - - -
P200 - - 108.4 124.6 48.2 65.3 47.0 34.0 66.9 71.3
wc 57.5 74.4 107.8 125.2 - - - - 57.6 86.7
LL - - - - 58.1 55.4 - - 78.6 60.1
PI 71.9 59.7 110.7 122.2 61.5 52.0 42.0 39.0 - -
%Saturation - - - - 36.1 77.5 20.0 61.0 - -
55
Table 4.12 Sensitivity Analysis (Effect of Response Variables on MR Prediction) Clay Soils # 4
MR+ = MR calculated with response variable increased by one standard deviation
MR- = MR calculated with response variable decreased by one standard deviation
1 MPa= 0.15 ksi
Table 4.13 Ranking of Response Variables Based on Sensitivity Sl. No. Explanatory Factor Significance* Acceptable Trend
1 Water content, wc 6/6 MR decreases with increase in Wc (5/6) 2 Percent saturation, S 4/4 MR decreases with increase in S (4/4) 3 Passing #200 sieve, P200 4/9 MR decreases with increase in P200 (7/9)
4 Plasticity Index, PI 3/8 MR decreases with decrease in PI? (5/8)
* If the change in MR exceeds 10% for a variation of one standard deviation
LTPP Carmichael Drumm Wyoming Mississippi Response Variable MR
+ MR - MR
+ MR - MR
+ MR - MR
+ MR - MR
+ MR -
%Clay 73.5 68.2 - - 75.8 70.7 - - - -
%Silt 74.6 67.2 - - - - - - - -
P40 72.7 68.9 - - - - - - - -
P200 65.9 76.1 81.3 99.1 63.9 82.6 53.0 40.0 93.3 97.4
wc 63.8 77.8 81.3 99.1 - - - - 76.9 123.9
LL 72.6 69.1 - - 74.9 71.5 - - 110.8 80.9
PI 70.9 70.7 78.5 101.9 82.9 63.6 48.0 44.0 - -
Saturation - - - - 52.4 94.1 26 67 - -
56
CHAPTER 5
SUMMARY AND CONCLUSIONS 5.1 Summary
The primary objective of this study is to validate and select a model for estimating
resilient modulus of soils for pavement design. The first category of models is heavily weighted
with soil index properties, and the second relies on stress state and soil index properties in
tandem. Eight Mississippi subgrade soils were tested in the laboratory for MR in accordance with
TP 46 protocol. Routine laboratory tests were also performed on the soils to determine the soil
index properties.
The predictability of the equations is sought by comparing predicted resilient modulus to
measured value and agreement thereof. Uncertainty in predicting resilient modulus arising from
inherent variability of soil physical properties is also studied, employing the point estimate
method. The significance of independent variables is investigated by conducting a sensitivity
study.
A critique of various equations investigated in this study reveals the following: Though
earlier equations emphasized soil index properties in predicting the resilient modulus, the current
trend is to start with a pseudo constitutive equation and then expand it to take into account soil
properties. Frequently used variables in developing the equations are moisture content, degree of
saturation, plasticity index, material passing the #200 sieve, and dry density. However, other
variables such as liquid limit, percent clay, percent silt, material passing the #40 sieve etc., are
also employed in a few equations.
57
5.2 Conclusions
The major conclusions resulting from the analysis validating the models are:
• Simple strength correlations, for example, the CBR test to estimate resilient modulus
should be used with caution.
• MR values predicted by the Georgia and Minnesota equations do not agree with the
laboratory values. Wyoming equation predictions are considered unsatisfactory as well.
• Carmichael and Drumm equations are not recommended, for the reason that estimation of
a few of the input parameters could be subjective and / or complex.
• Mississippi equations for fine-grain soil has resulted in close predictions in six out of
seven soils; and therefore, considered to be acceptable for predicting MR of Mississippi
soils. The coarse-grain soil equations, however, need to be revised.
• For having developed from an extensive materials database, LTPP equations have shown
potential to predict MR of a range of soils with a wide geographical coverage. In addition,
the models accommodate both stress variables and soil index properties in tandem. LTPP
models–coarse-grain, fine-grain silt and clay soils–therefore, deserve strong consideration
in level II Mechanistic Empirical pavement design.
Based on a sensitivity study of seven equations, investigating the significance of soil index
properties in predicting MR, the most important input variable is judged to be sample moisture
followed by material passing the #200 sieve, PI and sample density in that order.
5.3 Recommendation/Implementation of Results
As MDOT is in the process of implementing the Mechanistic Empirical Pavement Design
Guide, (ME PDG), subgrade characterization in terms of resilient modulus becomes a
58
prerequisite. Two sets of prediction equations deserve consideration for this purpose: LTPP
equations and Mississippi equations. Both models, in turn, present separate equations, one for
coarse soil and another for fine soil. For coarse soil (A-2 and A-3) LTPP equation 2.19 in
conjunction with 2.20 to 2.22 is the sole choice. For fine soil (A-4, A-5, A-6 and A-7), however,
the recommendation is to use both LTPP equations (2.23 to 2.28) and Mississippi equation 2.11,
and adopt an average of the two for design. Those values computed for both coarse- and fine-
grain soil could be used for a level II pavement design category or in preliminary design while
pursuing level I design. In the latter case, preliminary design could be revised when in-situ
resilient modulus becomes available, which can be ascertained only upon completion of the
grading project.
59
REFERENCES
1. AASHTO Guide for Design of Pavement structures, American Association of State Highway and Transportation Officials, Washington, DC, 1986.
2. NCHRP project 1-28A, Appendix A “Procedure for Resilient Modulus of
Unstabilized Aggregate Base and Subgrade Materials”, NCHRP Transportation Research Board, Washington D C, 2003.
3. AASHTO Guide for Design of Flexible pavement Structures, American Association
of State Highway and Transportation Officials, Appendix L Washington, DC, 1993. 4. Darter, M.I., R.P.Elliot, and K.T.Hall, “Revision of AASHTO pavement overlay
design procedures, Appendix: Documentation of Design Procedures”, National Cooperative Highway Research Program Study 20-7, TRB, National Research Council, Washington, D.C., April 1992.
5. Yau Amber, and H. L. Von Quintus., “Study of LTPP Laboratory Resilient Modulus
Test Data and Response Characteristics”, Final Report, October 2002 (FHWA-RD-02-051), USDOT, FHWA, October 2002.
6. Dai Shongtao, and John Zollars, “Resilient Modulus of Minnesota Road Research
Project Subgrade Soil”, Transportation Research Record 1786, TRB, National Research Council, Washington, D.C., 2002, pp 20-28.
7. Santha, B.L “Resilient Modulus of Subgrade Soils: Comparison of Two Constitutive
Equations”, Transportation Research Record 1462, TRB, National Research Council, Washington, D.C., pp 79-90.
8. Carmichael III R.F. and E. Stuart, “Predicting Resilient Modulus: A Study to
Determine the Mechanical Properties of Subgrade Soils”, Transportation Research Record 1043, TRB, National Research Council, Washington, D.C., 1978, pp 20-28.
9. Drumm E.C., Y. Boateng-Poku, and T. Johnson Pierce, “Estimation of Subgrade
Resilient Modulus from Standard Tests”, Journal of Geotechnical Engineering, Vol. 116, No.5, May 1900. pp 774-789.
10. Farrar, M J and J P Turner, “Resilient Modulus of Wyoming Subgrade Soils”,
Mountain Plains Consortium Report No 91-1, The University Of Wyoming, Luramie, Wyoming, 1991.
11. Rahim, A M and K. P. George, “Subgrade soil Index properties to Estimate Resilient Modulus”, Paper Presented at 83rd Annual Meeting of Transportation Research Board, Washington D C, 2004.
60
12. Yoder, E. J. and M. W. Witczak, Principles of Pavement Design, 2nd ed., John Wiley & Sons, New York, 1975.
13. Barksdale, R.D., Jorge, A., Khosla, N.P., Kim, K., Lambe, P.C., Rahman, M.S.,
“Laboratory Determination of Resilient Modulus for Flexible Pavement Design: Final Report (1998)”, NCHRP project 1-28.
14. Seed, S.B., Alavi, S.H., Ott, W.C., Mikhail, M., and Mactutis, J.A., “Evaluation of
Laboratory Determined and Nondestructive Test Based Resilient Modulus Values from WesTrack Experiment,” Nondestructive Testing of Pavements and Backcalculation of Moduli: Third Volume, ASTM STP 1375.
15. Li, D. and Ernest T. Seilg. “Resilient Modulus for Fine-Grained Subgrade Soils”.
Journal of Geotechnical Engineering, Vol. 120, No. 6, June, 1994. 16. Figueroa, J.L., and M.R. Thompson. Simplified Structural Analysis of Flexible
Pavements for Secondary Roads Based on ILLI-PAVE. Transportation Research Record 776, TRB, National Research Council, Washington, D.C., 1980, pp: 17-23.
17. Lekarp, F., U. Isacsson, and A., Dawson, “State of Art I: Resilient Response of
Unbound Aggregates”, Journal of Transportation Engineering, ASCE, Vol (1) Jan/Feb. 2000
18. Heukelom, W., and A.J.G. Klomp, “Dynamic Testing as a Means of Controlling
Pavements During and After Construction,” Proceedings of the First International Conference on Structural Design of Asphalt Pavements, University of Michigan, 1962.
19. Webb W.M., and B.E. Campbell, “Preliminary Investigation into Resilient Modulus
Testing for new AASHTO Pavement Design Guide,” Office of Materials and Research, Georgia Department of Transportation, 1986.
20. Thompson, M.R., and Q.L Robnett, “Resilient properties of subgrade soils.” Journal
of Transportation Engineering, ASCE, 105(1), 1976, 71-89. 21. Rada, G., and M. W. Witczak, “Comprehensive Evaluation of Laboratory Resilient
Moduli Results for Granular Materials”, Transportation Research Record 810, TRB, National Research Council, Washington, D.C., pp 23-33.
22. The Asphalt Institute, “Research and Development of the Asphalt Institute’s Thickness Design Manual, Ninth Edition,” Research Report No. 82-2, pp. 60-, 1982.
23. Yeh S.T., and C.K. Su, “Resilient Properties of Colorado Soils”, Colorado
Department of Highways, Report No. CDOH-DH-SM-89-9, 1989.
61
24. Dunlap, W.S, A report on a mathematical model describing the deformation characteristics of granular materials. Technical Report 1, Project 2-8-62-27, TTI, Texas A & M University, 1963.
25. Seed, H.B., F.G Mitry, C.L Monismith, and C. K. Chan, Prediction of pavement
deflection from laboratory repeated load tests, NCHRP Report 35, USA, 1967. 26. Uzan, J, “Characterization of Granular Material”, Transportation Research Record
1022, TRB, National Research Council, Washington, D.C., pp 52-59. 27. Brown, S.F., and J.W Pappin, “Analysis of Pavements with Granular Bases, Layered
Pavement Systems.” Transportation Research Record 810, TRB, National Research Council, Washington, D.C. 1981. pp: 17-23.
28. Moossazadeh J. and M. W. Witczak, “Prediction of Subgrade Moduli for Soil that
Exhibits Nonlinear Behavior”, Transportation Research Record 810, TRB, National Research Council, Washington, D.C., pp 9-17.
29. May, R. W. and M. W Witczak, “Effective Granular Modulus to Model Pavement
Response”. Transportation Research Record 810, TRB, National Research Council, Washington, D.C. 1981
30. “Unbound Materials Characterization Utilizing LTPP Data,” Unpublished Interim
Task Report Project 1-37A, Development of the 2002 Design Guide for the Design of New and Rehabilitated Pavement Structures, National Cooperative Highway Research Program, National Research Council, Washington D. C., August 2000.
31. Stubstad, R.N. et al., “NCHRP Web Document # 52: LTPP Data Analysis: Feasibility
of Using FWD Deflection Data to Characterize Pavement Construction Quality”, Project 20-59 (9), Washington, D.C., 2002.
32. Von Quintus, H. and B. Killingsworth, “Analyses Relating to Pavement Material
Characterizations and Their Effects on Pavement Performance,” Report No. FHWA-RD-97-085, Federal Highway Administration, U.S. Department of Transportation, Washington, D.C., 1998.
33. Mohammad L. N., B. Huang, A J. Puppala, and A. Allen, “Regression Model for Resilient Modulus of Subgrade Soils”, Transportation Research Record 1442, TRB, National Research Council, Washington, D.C., pp 47-54.
34. Kyalham, V. and M. Willis, “ Predictive Equations for Determination of Resilient
Modulus, Suitability of Using California Bearing Ratio Test to Predict Resilient Modulus”, presented to Federal Aviation Administration, 2001.
35. George, K.P., “Falling Weight Deflectometer For Estimating Subgrade Resilient
Moduli,” Final Report, Submitted to Mississippi Department of Transportation, University of Mississippi, July 2003.
62
36. Elliot, R. P. “Selection of Subgrade Modulus for AASHTO Flexible Pavement
Design”, Transportation Research Record 1354, TRB, National Research Council, Washington, D.C., 1992.
37. Huang, Y.H. Pavement Analysis and Design, by Prentice-Hall, Inc., New Jersey,
1993. 38. Rosenblueth, E (1975), “Point Estimates for probability moments”, proceedings of
the National Academy of Science, USA, Vol. 72, No.10, pp 3812-3814. 39. Lee, I.K., White, W., and Ingles, O.G., Geotechnical Engineering, Pitman Publishing
Inc, Massachusetts, 1983, ISBN 0-273-01756-X.
63
APPENDIX A Testing Sequence for Subgrade Soil Materials, TP 46 Protocol
Confining Pressure, σ3
Seating Stress, 0.1 σmax
Cyclic Stress, σcyclic
Max. Axial Stress, σmax
Sequence No.
psi psi psi psi
No. of Load Applications
0 6 0.4 3.6 4 500-1000
1 6 0.2 1.8 2 100
2 6 0.4 3.6 4 100
3 6 0.6 5.4 6 100
4 6 0.8 7.2 8 100
5 6 1.0 9.0 10 100
6 4 0.2 1.8 2 100
7 4 0.4 3.6 4 100
8 4 0.6 5.4 6 100
9 4 0.8 7.2 8 100
10 4 1.0 9.0 10 100
11 2 0.2 1.8 2 100
12 2 0.4 3.6 4 100
13 2 0.6 5.4 6 100
14 2 0.8 7.2 8 100
15 2 1.0 9.0 10 100
64
APPENDIX B Method of Point Estimates: Illustration
Let Y be a function of two random variables, Y= f(x1, x2)
Approximate values of the first two moments of a function (Y) from the first two moments of the
random variables (x1, x2) can be obtained from the method of point estimates.
The mean and variance of Y are given by,
Mean, µY = P++Y++ + P+-Y+- + P-+Y-+ + P- -Y- -
Variance, σ2Y =P++Y2
++ + P+-Y2+- + P-+Y2
-+ + P- -Y2- - - µ2
Y
where, Y++ = Y (x1+, x2+);
Y+- = Y (x1+, x2- );
Y-+ = Y (x1-, x2+ );
Y-- = Y (x1-, x2- );
x+ = µx + σx;
x- = µx - σx;
P++ = P- - = 0.25 (1 + rx1, x2);
P+ - = P- + = 0.25 (1 - rx1, x2); and
rx1,x2 is the correlation coefficient between x1, x2. If x1, x2 are independent,
then rx1,x2 = 0, an assumption adopted in all of the calculations.
top related